We have a standard one that we send out with our bank reference and trade vendor information on. Accounts Payable enters the vendors in the system, but Procurement has to approve all new vendors prior to doing business with them. In this case, whose responsibility would it be to fill out the credit application if the vendor will not accept our filled out credit application, AP or Procurement?
There is no right or wrong approach. This may be handled by various departments, depending on how the company is organized.
The best practice should be based on a process which protects and ensures efficiency and protection of this data. It should be handled by only one department, whether you send out a pre-filled trade reference sheet or have to fill out the vendor's credit application form. The vendor application will usually require the same information you have on your trade reference form (which you call a credit application), and possibly a few extra data fields. The vendor may insist on using their form because it might be used to scan the information into their system.
Since there seems to be a stand-off on who should do it, I would suggest mapping the process. Rather than having a back-and-forth, design a new process with fewer touch points, controlled by one department that sends out the trade/bank references or fills out the vendors' credit application. Make sure to build in controls that ensure security of this sensitive information and in which there is formal approval, so the vendor master data entry team knows they can set up a new vendor.